On Some Problems for Doubly Nonlinear Parabolic Equations and Equations of Variable Type

In the article, existence theorems are proven of regular solutions to the first and second boundary value problems for a wide class of parabolic equations with double nonlinearity which may degenerate on a solution and on the gradient of a solution. Solvability of variational inequalities connected with these equations is studied. The existence of $t$-periodic solutions to nonlinear parabolic equations with changing time direction is established.